iccv iccv2013 iccv2013-183 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Wei Zeng, Mayank Goswami, Feng Luo, Xianfeng Gu
Abstract: Surface registration plays a fundamental role in many applications in computer vision and aims at finding a oneto-one correspondence between surfaces. Conformal mapping based surface registration methods conformally map 2D/3D surfaces onto 2D canonical domains and perform the matching on the 2D plane. This registration framework reduces dimensionality, and the result is intrinsic to Riemannian metric and invariant under isometric deformation. However, conformal mapping will be affected by inconsistent boundaries and non-isometric deformations of surfaces. In this work, we quantify the effects of boundary variation and non-isometric deformation to conformal mappings, and give the theoretical upper bounds for the distortions of conformal mappings under these two factors. Besides giving the thorough theoretical proofs of the theorems, we verified them by concrete experiments using 3D human facial scans with dynamic expressions and varying boundaries. Furthermore, we used the distortion estimates for reducing search range in feature matching of surface registration applications. The experimental results are consistent with the theoreticalpredictions and also demonstrate the performance improvements in feature tracking.
Reference: text
sentIndex sentText sentNum sentScore
1 Conformal mapping based surface registration methods conformally map 2D/3D surfaces onto 2D canonical domains and perform the matching on the 2D plane. [sent-6, score-0.723]
2 This registration framework reduces dimensionality, and the result is intrinsic to Riemannian metric and invariant under isometric deformation. [sent-7, score-0.263]
3 However, conformal mapping will be affected by inconsistent boundaries and non-isometric deformations of surfaces. [sent-8, score-0.807]
4 In this work, we quantify the effects of boundary variation and non-isometric deformation to conformal mappings, and give the theoretical upper bounds for the distortions of conformal mappings under these two factors. [sent-9, score-1.793]
5 Besides giving the thorough theoretical proofs of the theorems, we verified them by concrete experiments using 3D human facial scans with dynamic expressions and varying boundaries. [sent-10, score-0.252]
6 Furthermore, we used the distortion estimates for reducing search range in feature matching of surface registration applications. [sent-11, score-0.65]
7 3D surface registration is a fundamental tool, which aims at finding a mapping (one-to-one correspondence) between two 3D surfaces. [sent-16, score-0.467]
8 3D surface tracking requires generating the mappings over a series of surface frames in 3D spatial-temporal data (3D geometry video or 4D data). [sent-17, score-0.585]
9 In the last decade, many 3D surface registration methods have been developed. [sent-24, score-0.366]
10 The most well-known iterative closest point (ICP) method [2] works well for rigid motions but cannot handle the nonrigid deformation and inconsistent boundary problems. [sent-25, score-0.303]
11 Surface conformal mapping based methods have been developed for surface matching [17, 6, 12], registration [3, 25, 26], and tracking [27]. [sent-27, score-1.085]
12 The key idea is to map surfaces to 2D canonical domains and then solve the surface registration problem as an image registration problem. [sent-28, score-0.658]
13 The conformal mapping based methods can handle nonrigid deformations and generate diffeomorphisms between surfaces. [sent-29, score-0.757]
14 Surface conformal mapping can be generalized to surface quasiconformal mapping, which has great potential to handle large-scale nonrigid (including non-isometric) deformations in surface registration application [24, 13, 14]. [sent-30, score-1.627]
15 The fundamental questions to be addressed in this work are as follows: How to quantify the effects of boundary variation and nonisometric deformation to the conformal mapping? [sent-33, score-0.891]
16 How sensitive is the conformal mapping based registration method to boundary variation and non-isometric deformation? [sent-34, score-1.024]
17 Therefore, when we map two adjacent frames onto the unit disk by conformal mappings, the deviation of the two images of the same point on the face surface is expected to be small. [sent-40, score-0.978]
18 It is highly desirable to quantify the deviation on conformal mapping in terms of both boundary variation and non-isometric defor- mation. [sent-41, score-0.875]
19 Furthermore, once we have the distortion estimation of conformal maps, we can reduce the search range for matching the feature points of two surfaces, which will improve the efficiency and accuracy of surface registration. [sent-42, score-1.042]
20 This work presents upper bound estimation on distortions of conformal maps caused by surface boundary variations and non-isometric deformations. [sent-43, score-1.082]
21 These theoretical results are not classical, and to the best of our knowledge, this is the first work to give rigorous theoretical upper bound estimation on distortion ofconformal maps in the above cases. [sent-44, score-0.43]
22 The framework of surface registration based on conformal mapping is as follows. [sent-46, score-1.044]
23 First we conformally map both of them to the unit disk φk : Sk → D, k = 1,2. [sent-48, score-0.3]
24 If we can find a mapping between these two planar images φ˜ : D → D, then the registration can be obtained as φ = ◦ φ˜ ◦ φ1 . [sent-49, score-0.286]
25 The angle distortion K of a quasiconformal map is defined as − K = ? [sent-60, score-0.559]
26 Conformal map is a special case of quasiconformal map whose μ is zero everywhere. [sent-64, score-0.323]
27 Our task is to estimate the upper bound of the differences between two images of the same point p ∈ Ω in terms of both boundary variations ε1 ,ε2 and the angle distortion K, |φ1 (p) − φ2(p) | < g(ε1 ,ε2, K) ,∀p ∈ Ω. [sent-67, score-0.466]
28 By applying geometric methods from quasiconformal geometry and harmonic analysis, we give the explicit upper bound g(ε1 ,ε2, K), which is a linear function of both εk’s and K (see Theorems 3. [sent-68, score-0.455]
29 In this work, we estimate the distortions of conformal mappings both theoretically and experimentally. [sent-73, score-0.819]
30 We present three main theorems, along with rigorous proofs based on quasiconformal geometry, which state the upper bounds of distortion with respect to the boundary variation and non-isometric deformation. [sent-74, score-0.817]
31 We then utilize the distortion bound for feature registration purposes. [sent-75, score-0.462]
32 Quantify the distortion of conformal mappings caused by boundary variation. [sent-78, score-1.134]
33 Quantify the distortion of conformal mappings caused by non-isometric deformations. [sent-80, score-1.003]
34 Present a registration algorithm which searches corre- sponding features in the given distortion range. [sent-82, score-0.399]
35 The quasiconformal mapping is uniquely determined by μ up to a conformal transformation, which is stated in the classical measurable Riemann mapping theorem [1]. [sent-99, score-1.254]
36 Then there exists a quasiconformal homeomorphism f : → D, whose Beltrami coefficient is μ. [sent-105, score-0.436]
37 Ω Namely, the space of quasiconformal homeomorphism (QCH) between and D and the functional space of Beltrami coefficients have the following relation Ω QCH(S,D)/{M o¨bius} ∼= {μ|μ : S → C, ? [sent-107, score-0.362]
38 It conformally deforms the Riemannian metric and converges to constant curvature metric [5, 10]. [sent-114, score-0.286]
39 Then, depending on the topology of S, it can be conformally deformed to one of three canonical shapes: the unit sphere S2, the flat torus R2/Γ (Γ is a subgroup of Euclidean translation), or H2/Γ (Γ is a subgroup of hyperbolic rigid motion group). [sent-117, score-0.239]
40 This shows that for surfaces with Riemannian metrics, we can find a conformal atlas {(Uα, φα)}, such that all the chart transitions φαβ : φα (Uα Uβ ) → φβ (Uα Uβ ) are the elements in Γ, which are planar conformal mappings. [sent-118, score-1.261]
41 The pullback metric induced by φ on Ω is φ∗|dw|2 = |wz|2|dz+μd z¯|2, therefore, the pullback metric is conformal to the auxiliary metric |dz + μd¯ z|2. [sent-124, score-0.882]
42 3 (Auxiliary Metric) A quasiconformal mapping associated with Beltrami coefficient μ becomes a conformal mapping under the auxiliary metric, φ : (Ω, |dz|2,μ) → (D, |dw|2) ≡ φ : (Ω, |dz+μd¯ z|2) → (D, |dw|2). [sent-126, score-1.233]
43 We use concepts from harmonic measure [9] and quasiconformal geometry [1] to obtain the distortion bounds. [sent-129, score-0.57]
44 Boundary Variation We first estimate the distortion arising due to inconsistent boundaries. [sent-132, score-0.262]
45 The conformal mappings of the scanned static faces might be almost identical except that their boundaries do not really match. [sent-133, score-0.795]
46 One then needs to estimate how much the conformal mapping of one surface differs from another. [sent-134, score-0.859]
47 In the first case the boundary of one surface is contained in the other. [sent-136, score-0.312]
48 Let φ : −→ D be the Riemann mapping which fixes the origin and (the continuous extension of φ to the boundary of Ω) maps the point p = Γ ∩ R+ (note that star-shaped property of Γ implies uniqueness of p) to the point 1. [sent-144, score-0.256]
49 The proof is based on harmonic measure and its invariance under conformal changes. [sent-152, score-0.61]
50 Distortion estimation of conformal mappings between two surfaces S1,S2 with big overlapping region Ω. [sent-159, score-0.864]
51 Non-isometric Deformation We then estimate the magnitude and argument changes for a point inside a domain under the action of a quasiconformal map, usually induced by a non-isometric deformation. [sent-181, score-0.375]
52 As previously mentioned, quasiconformal maps intuitively distort angles by a bounded amount. [sent-182, score-0.323]
53 This distortion is encoded in the Beltrami coefficient μ of the map. [sent-183, score-0.288]
54 By the Riemann mapping theorem, all simply connected domains are conformally equivalent to the unit disk D. [sent-184, score-0.401]
55 without loss of generality, we just study the distortion caused by a quasiconformal mapping on the unit disk. [sent-185, score-0.726]
56 Assume that all quasiconformal mappings f : D → D are normalized, i. [sent-186, score-0.503]
57 The distortion estimation is based on the deformation theorem for quasiconformal mappings [1]. [sent-197, score-0.921]
58 The bound of the deformation velocity field |V(t, z) | can be estimated, whose integration along t gives the bound of the distortion |h(1,z) −h(0, z) |, namely, |f(z) −z| . [sent-200, score-0.428]
59 The discrete conformal factor is defined as a function λ : V → R. [sent-210, score-0.599]
60 Quasiconformal Mapping by Auxiliary Metric We apply the quasiconformal mapping algorithm based on the auxiliary metric of Theorem 2. [sent-220, score-0.538]
61 The discrete surface Ricci flow algorithm takes the Riemannian metric as input. [sent-223, score-0.292]
62 Here we use the auxiliary metric to replace the original induced Euclidean metric of the surface, then computing a quasiconformal mapping becomes computing a conformal mapping. [sent-224, score-1.197]
63 Algorithm 1Auxiliary Metric Input: Triangular mesh M = (V,E,F) with conformal parameterization z : V → C, and discrete Beltrami coefficient μ : V → C defined on the conformal structure. [sent-225, score-1.27]
64 This work gives the upper theoretical bound for the search range (distortion) estimation in terms of two parameters: 1) the boundary difference represented by ε; 2) the geometric difference represented by μ. [sent-233, score-0.328]
65 In this section, we verify the distortion theorems and evaluate the performance for dynamic feature registration and tracking application by running experiments on 3D human facial scans with dynamic expressions and inconsistent boundaries. [sent-236, score-0.762]
66 If we fix the subject and change the expression, then the physical real mapping between the facial surfaces can be fully recovered by its local angle distortion μ; if we change the subject, then the best matching between two faces also gives the μ. [sent-238, score-0.531]
67 Then, we apply the distortion upper bounds for feature registration using the boundary variation and non-isometric deformation parameters obtained from our experiments. [sent-257, score-0.714]
68 Static Surface with Boundary Variation Figure 3 shows the input face surface (a), which is conformally mapped to the planar unit disk (b). [sent-260, score-0.533]
69 We conformally map all Sk’s to the unit disk with normalization condition, φk : Sk → D. [sent-263, score-0.3]
70 The measurements of all the boundary points on conformal mappings are plotted in (d), which is consistent with the distortion estimation theorem on boundary containment (see Theorem 3. [sent-266, score-1.412]
71 In addition, Figure 4 illustrates the conformal mappings in the top row, and shows the deformation vector fields φk(zj) φ0(zj) of a set of feature points zj ∈ Sk in the bottom row. [sent-268, score-0.953]
72 Conformal mappings of a 3D human facial surface with boundary variations. [sent-277, score-0.561]
73 The original boundary of surface (a) is mapped to the red circle on the unit disk (b). [sent-278, score-0.525]
74 (d) plots the distortion of conformal mapping with respect to boundary deviation ε. [sent-280, score-1.023]
75 The upper row displays the conformal mappings for S1 , S3 , S5 ,S9; the bottom row illustrates the deformation vector fields of feature points plotted on the original conformal mapping (see Fig. [sent-286, score-1.583]
76 First, we conformally map the face surface to the unit disk (b). [sent-292, score-0.512]
77 2 to get the quasiconformal mappings as shown in (c-d). [sent-294, score-0.503]
78 The distortion curve is plotted in (e), which is consistent with the distortion estimation theorem on non-isometric deformation (see Theorem 3. [sent-296, score-0.656]
79 We track the face sequence using the algorithm in [24], then compute the Beltrami coefficients for each Sk using S0 as the reference surface, fk : S0 → Sk, and measure the distortion maxz∈D |fk(z) −z| . [sent-301, score-0.245]
80 Dynamic Surfaces with Inconsistent Boundary We capture the dynamic facial surface sequence {Sk} with both expression and pose changes (scanning speed is 30 fps [21]), as shown in Fig. [sent-308, score-0.344]
81 The facial surface sequence obtained includes both inconsistent boundaries and non-isometric expression deformations (the boundary changes are caused by head rotation, and human facial expressions are non-isometric [23]). [sent-310, score-0.698]
82 We then compute the normalized conformal mappings φk : Sk → D as described 2637 (a)3Dsurface(b)μ0(z)=0(c)μ0. [sent-311, score-0.757]
83 Quasiconformal mappings between a human face surface and a unit disk with varying μ. [sent-323, score-0.561]
84 Conformal mappings for a human facial expression sequence {Sk}. [sent-325, score-0.306]
85 The sparse feature points on face Sk are computed first [23]; the generated correspondences are used as ground truth for evaluating the registration accuracy of our feature registration method. [sent-343, score-0.401]
86 Frames (a) and (b) show two facial scans, with big boundary change and a nonisometric expression change, (c) shows the search range for all feature points on the source surface, and (d) shows the displacements of the matched feature points. [sent-345, score-0.338]
87 By using the proposed distortion estimation, the efficiency for matching has been improved by 5 times, compared with a conformal mapping based registration method using heuristic search range [27]. [sent-349, score-1.147]
88 For dynamic facial surfaces with expressions, if the scanning speed is fast, then the norm bound is small; if face expression matching has been done many times, then the norm bound can be learned. [sent-353, score-0.532]
89 To the best of our knowledge, till today, this is the on- ly work which gives the distortion estimations of conformal mappings with respect to the variation of boundary and the norm bound of Beltrami coefficient, along with rigorous mathematical proofs. [sent-358, score-1.261]
90 We have tested a broader range of examples on 4 facial sequences with various expression and pose changes (each sequence has 400 frames) and sampled totally 200 pairs of surfaces for computing the distortions of conformal mappings; the experimental results validate the theoretical results. [sent-359, score-0.943]
91 Conclusion The accuracy and stability of surface registration based on conformal mapping method depends on the boundary consistency and the non-isometric deformation. [sent-361, score-1.175]
92 Conformal mapping changes when both the boundary and expression vary. [sent-363, score-0.289]
93 (c) shows the search range for each feature point of (a), where the red and blue dots denote the feature points of (a) and (b), respectively; the yellow circle illustrates the search range for each feature point of (a) on conformal mapping domain (a unit disk) with the search radius 0. [sent-365, score-0.888]
94 work, we estimate the distortions of conformal mappings theoretically and experimentally. [sent-368, score-0.819]
95 We quantify the distortions of conformal mappings caused by both boundary variation and non-isometric deformations. [sent-369, score-1.048]
96 The main theorems claim that the distortions are linear functions of the upper bound of the boundary variation and non-isometric deformation parameters. [sent-370, score-0.487]
97 Furthermore, we present a feature registration algorithm which searches the corresponding features within the range given by the distortion estimation, which improves the efficiency and accuracy of registration. [sent-371, score-0.424]
98 In future work, we will further improve the bounds of the distortion estimation, and explore further surface tracking algorithm utilizing the theoretical results of this work. [sent-372, score-0.494]
99 3D non-rigid surface matching and registration based on holomorphic differentials. [sent-555, score-0.41]
100 Dense non-rigid surface registration using high-order graph matching. [sent-564, score-0.366]
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